Computational Chemistry 4.25 - Fock Matrix

Short lecture on the Fock matrix in Hartree-Fock theory.

The Fock operator is the core Hamiltonian operator (electron kinetic energy plus its attraction to all nuclei) plus a sum over all spatial orbitals of twice the Coulomb operator subtracted by the exchange operator. The Fock matrix is a K by K matrix composed of elements which are the expectation value of the Fock operator acting on an atomic orbital basis function and a different basis function complex conjugate. This is equivalently the core Hamiltonian matrix added to a sum of two-electron integrals. The core Hamiltonian matrix (H) is a sum of the kinetic energy (T) and nuclear potential (V) matrices. The two-electron part of the Fock operator can be refactored to be orbital invariant with the G matrix, which is a double sum over all basis functions of the a density matrix (P) element and the difference of two two-electron integrals. Using the Hartree-Fock-Roothaan equations, we can find the orbital coefficient matrix (C) by diagonalizing the Fock matrix to find these eigenvectors.

Notes Slide: https://i.imgur.com/JNx53eh.png

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